2 Nov
2020
2 Nov
'20
7:42 p.m.
If this "d-to-1 map" h : X β> Y is a covering map, then the covering space (the domain) will have d times the number of vertices, edges, and faces of the image. This means π(X) = d π(Y). I presume we're talking about compact orientable surfaces. Since for them, π = 2 - 2g, this tells us that 2 - 2g_X = d (2 - 2g_Y), or g_X = d g_Y - d + 1 βDan Jim Propp wrote: ----- What is the relation between the genus of X and the genus of Y when there is a d-to-1 map from X to Y? (Assume that around each y in Y we can find a disk whose preimage consists of d disks.) Do we have genus(X) = d genus(Y) ? -----